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The Nonexistence of a []-Quantum stabilizer code

IEEE TRANSACTIONS ON INFORMATION THEORY 2011年第7期57卷 4788-4793页

作者： Bierbrauer, Juergen Fears, Richard Marcugini, Stefano Pambianco, Fernanda Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA Univ Perugia Dipartimento Matemat & Informat I-06100 Perugia Italy

One of the oldest problems in the theory of quantum stabilizer codes is solved by proving the nonexistence of quantum [[13, 5, 4]]-codes.

关键词： Additive code codeline dimension hyperplane quantum code stabilizer code strength symplectic geometry weight

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On the construction of stabilizer codes with an arbitrary binary matrix

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QUANTUM INFORMATION PROCESSING 2013年第1期12卷 467-479页

作者： Hwang, Yongsoo Choi, Byung-Soo Jeon, Moongu Gwangju Inst Sci & Technol Dept Informat & Commun Kwangju South Korea Duke Univ Dept Elect & Comp Engn Durham NC USA

This paper proposes a simple framework for constructing a stabilizer code with an arbitrary binary matrix. We define a relation between A (1) and A (2) of a binary check matrix A = (A (1)|A (2)) associated with stabilizer generators of a quantum error-correcting code. Given an arbitrary binary matrix, we can derive a pair of A (1) and A (2) by the relation. As examples, we illustrate two kinds of stabilizer codes: quantum LDPC codes and quantum convolutional codes. By the nature of the proposed framework, the stabilizer codes covered in this paper belong to general stabilizer (non-CSS) codes.

关键词： Quantum error-correcting code stabilizer code Low-density parity check code Convolutional code

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Measures of 7-qubit stabilizer codes for graph states 4

Measures of 7-qubit stabilizer codes for graph states

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4th International Conference on Intelligent System and Applied Material (GSAM)

作者： Wang, Yunyun Jiang, Lizhen Gongshang Univ Coll Informat & Elect Engn Hangzhou 310018 Zhejiang Peoples R China

ISBN: (纸本)9783038353140

Most quantum error-correcting codes constructed are stabilizer codes which are potentially more efficient and significant. Under the concept of graph states theory, we focused on 7-qubit graph states characteristics of stabilizer codes, by calculating and analyzing the entanglement properties of graphical codes. For the instance of code ((7,1,3)) with 16 inequitable graphs, the figure of entanglement properties can be measured. And we analyzed 7-qubit graph characteristics of stabilizers codes and calculated the entanglement measures by iterative algorithm.

关键词： stabilizer code Graph state Iterative algorithm

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High Performance Entanglement-Assisted Quantum LDPC codes Need Little Entanglement

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IEEE TRANSACTIONS ON INFORMATION THEORY 2011年第3期57卷 1761-1769页

作者： Hsieh, Min-Hsiu Yen, Wen-Tai Hsu, Li-Yi Japan Sci & Technol Agcy ERATO SORST Quantum Computat & Informat Project Bunkyo Ku Tokyo Japan Chung Yuan Christian Univ Dept Phys Chungli 32023 Taiwan

Though the entanglement-assisted formalism provides a universal connection between a classical linear code and an entanglement-assisted quantum error-correcting code (EAQECC), the issue of maintaining large amount of pure maximally entangled states in constructing EAQECCs is a practical obstacle to its use. It is also conjectured that the power of entanglement-assisted formalism to convert those good classical codes comes from massive consumption of maximally entangled states. We show that the above conjecture is wrong by providing families of EAQECCs with an entanglement consumption rate that diminishes linearly as a function of the code length. Notably, two families of EAQECCs constructed in the paper require only one copy of maximally entangled state no matter how large the code length is. These families of EAQECCs that are constructed from classical finite geometric LDPC codes perform very well according to our numerical simulations. Our work indicates that EAQECCs are not only theoretically interesting, but also physically implementable. Finally, these high performance entanglement-assisted LDPC codes with low entanglement consumption rates allow one to construct high-performance standard QECCs with very similar parameters.

关键词： Low density parity check codes Euclidean geometry projective geometry cyclic code stabilizer code entanglement-assisted code quasi-cyclic code

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Advance Sharing of Quantum Shares for Quantum Secrets

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2024年第8期E107A卷 1247-1254页

作者： Shibata, Mamoru Matsumoto, Ryutaroh Tokyo Inst Technol Dept Informat & Commun Enginnering Tokyo 1528550 Japan

Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, , 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.

关键词： quantum secret sharing advance sharing stabilizer code EAQECC

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Lecture notes on quantum entanglement:From stabilizer states to stabilizer channels

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Frontiers of physics 2024年第5期19卷 207-226页

作者： Amir R.Arab Steklov Mathematical Institute of Russian Academy of Sciences Gubkina Str.8Moscow 119991Russia Moscow Institute of Physics and Technology 9 Institutskiy Per.DolgoprudnyMoscow Region141701Russia

We study mathematical,physical and computational aspects of the stabilizer formalism arising in quantum information and quantum *** measurement process of Pauli observables with its algorithm is *** is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ(Greenberger-Horne-Zeilinger)*** discuss stabilizer codes and construct a stabilizer code from a given linear *** also discuss quantum error correction,error recovery criteria and syndrome *** symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial *** structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer *** distance of embeddable stabilizer codes in lattices is *** discuss the Knill-Gottesman theorem,tableau representation and frame *** runtime of simulating stabilizer gates is obtained by applying stabilizer ***,an algorithm for updating global phases is *** of quantum channels into stabilizer channels is *** discuss capacity achieving codes to obtain the capacity of the quantum erasure ***,we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.

关键词： Pauli product stabilizer state measurement process entanglement detection stabilizer code stabilizer circuit quantum channel tomography

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Partially Concatenated Calderbank-Shor-Steane codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第1期69卷 262-272页

作者： Fan, Jihao Li, Jun Wang, Ya Li, Yonghui Hsieh, Min-Hsiu Du, Jiangfeng Nanjing Univ Sci & Technol Sch Cyber Sci & Engn Nanjing 210094 Peoples R China Nanjing Univ Sci & Technol Sch Elect & Opt Engn Nanjing 210094 Peoples R China Univ Sci & Technol China CAS Key Lab Microscale Magnet Resonance Hefei 230026 Peoples R China Univ Sci & Technol China Dept Modern Phys Hefei Peoples R China Univ Sydney Sch Elect & Informat Engn Sydney NSW 2006 Australia Hon Hai Res Inst Quantum Comp Res Ctr Taipei 114 Taiwan

In this paper, we utilize a concatenation schemeto construct new families of quantum error correction codesachieving the quantum Gilbert-Varshamov (GV) bound asymp-totically. Weconcatenate alternant codes with any linear codeachievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code canachieve the quantum GV bound asymptotically and can approachthe Hashing bound for asymmetric Pauli channels. By combingSteane's enlargement construction of CSS codes, we derive afamily of enlarged stabilizer codes achieving the quantum GVbound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parametersQ1=[[N,omega(root N),omega(root N)]],andQ(2)=[[N,omega(N/log N),omega(N/logN)/omega(log N)]].We show thatQ1can be encoded very efficiently by circuits of sizeO(N)and depth O(root N). For an input error syndrome,Q1cancorrect any adversarial error of weight up to half the minimum distance bound in O(N)time.Q1can also be decoded in parallelinO(root N)time by using O(root N) classical processors. For an input error syndrome, we proved thatQ2can correct a linear number of X-errors with high probability and an almostlinear number of Z-errors in O(N)time. Moreover,Q2canbe decoded in parallel inO(log(N))time by usingO(N)classical processors

关键词： codes Linear codes Encoding Decoding Program processors Task analysis Qubit stabilizer code Calderbank-Shor-Steane code asymmetric quantum code quantum Gilbert-Varshamov bound Pauli channel

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Unitary reconstruction of secret for stabilizer-based quantum secret sharing

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QUANTUM INFORMATION PROCESSING 2017年第8期16卷 1-9页

作者： Matsumoto, Ryutaroh Nagoya Univ Dept Informat & Commun Engn Nagoya Aichi 4648603 Japan

We propose a unitary procedure to reconstruct quantum secret for a quantum secret sharing scheme constructed from stabilizer quantum error-correcting codes. Erasure correcting procedures for stabilizer codes need to add missing shares for reconstruction of quantum secret, while unitary reconstruction procedures for certain class of quantum secret sharing are known to work without adding missing shares. The proposed procedure also works without adding missing shares.

关键词： Quantum secret sharing Quantum error correction stabilizer code

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Fractons

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ANNUAL REVIEW OF CONDENSED MATTER PHYSICS, VOL 10 2019年第1期10卷 295-313页

作者： Nandkishore, Rahul M. Hermele, Michael Univ Colorado Dept Phys Boulder CO 80309 USA Univ Colorado Ctr Theory Quantum Matter Boulder CO 80309 USA

Fracton phases constitute a new class of quantum state of matter. They are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics or mobile only in certain directions. These phases do not wholly fit into any of the existing paradigms but connect to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective and allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at nonzero density before concluding with a brief survey of some open problems.

关键词： stabilizer code glassy dynamics spin liquid symmetric tensor entanglement

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A Characterization of Entanglement-Assisted Quantum Low-Density Parity-Check codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2013年第6期59卷 3347-3353页

作者： Fujiwara, Yuichiro Tonchev, Vladimir D. CALTECH Div Phys Math & Astron Pasadena CA 91125 USA Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA

As in classical coding theory, quantum analogs of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.

关键词： Entanglement-assisted quantum error correction low-density parity-check (LDPC) code pairwise balanced design stabilizer code Steiner 2-design

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